Generally, I think the case for each side of the argument is somewhat overstated. While I don't think UG provides an significant boost, I also don't think it's a negligible one. Numbers are numbers regardless of where they come from, but there are few things that have led me to believe that UG has a bigger impact that many TLS posters let on. That's my view based on first-hand knowledge and some hearsay (which I've come to believe):
1. I just graduated from a (quote unquote) Top 10 school and having met with our pre-law advisor a few times, I do think that being from one of these schools can give applicants a noticeable bump in admissions. The advisor gave us the average LSAT/GPA combos for applicants from my undergrad who were accepted to T14 schools, and for nearly all of them, the average LSAT, GPA, or both were slightly below those law schools' published medians (E.g. admitted avg GPA to HLS from my school was more like a 3.8 than a 3.9, etc.). The advisor gave me averages, which I compared to medians, so that could have skewed it somewhat. Also, the advisor seemed to have formed relationships with many of these schools' adcomms, which I would guess would help in instances where applicants push hard, with the help of the advisor, to get off the waitlist. I wouldn't be surprised if that occurs given how frequent it is in undergrad admissions (the book The Gatekeepers illustrates just how much sway the college counselors at prestigious prep schools have at getting their students into top UGs, and I'd think the same dynamics could come into play with law school).
2. The other thing I've heard is that UG prestige can help with getting that first 1L summer job, but thereafter, the difference of UG has no discernible effect on career prospects. Not sure if that's true or not.
But if it is, I could see adcomms maybe giving an edge to certain UG degrees.
3. I have no idea if this is a thing, but I could also see adcomms preferring a 3.7 from princeton to a 3.7 from JohnDoe given the differences in LSAT averages at that school. If the princeton average LSAT is a 165, and the JohnDoe average is a 153, it might suggest that the Princeton student had greater competition to secure that GPA because of his/her talented classmates (obviously, this is assuming both schools grade on a similar curve, which might not be a reasonable assumption).
Just my 2 cents. Some evidence, even more speculation